⚛️ phenomenology

Entropy and DIS structure functions

This paper proposes an improved method for determining entanglement entropy in Deep Inelastic Scattering (DIS) using proton structure functions and parton distribution functions, demonstrating that the results align closely with H1 experimental data across various kinematic ranges.

Original authors: G. R. Boroun

Published 2026-02-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: G. R. Boroun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a massive, incredibly complex jigsaw puzzle of a galaxy, but you are only allowed to look through a tiny, high-powered magnifying glass. You can only see a few pieces at a time. This is essentially what physicists are doing when they study the proton—the tiny building block inside every atom.

This paper, written by G.R. Boroun, explores a concept called "Entanglement Entropy" to understand how much "information" is hidden inside that proton.

Here is a breakdown of the paper using everyday analogies.

1. The Concept: The "Hidden Party" (Entanglement Entropy)

Imagine you are at a massive, crowded music festival. You are standing in a small VIP section (this is the "Region A" the scientists can see). You see a few people dancing and drinking soda. However, you know that just outside your VIP fence, there is a massive crowd of thousands of people (this is "Region B").

Even though you can't see the crowd outside, the fact that they are all part of the same festival means they are "connected" to the people in your VIP section. If the music changes, everyone reacts. In physics, this connection is called Entanglement.

Entropy is a measure of how much "uncertainty" or "information" there is. If you only see a few people in the VIP section, you have no idea how wild the party is outside. The "Entanglement Entropy" tells us: "Based on the few pieces I can see here, how much chaos and complexity must be happening in the part I can't see?"

2. The Problem: The "Blurry Map" (PDFs vs. Structure Functions)

To study the proton, scientists usually use something called PDFs (Parton Distribution Functions). Think of a PDF as a map that tells you where the "parts" (quarks and gluons) are located inside the proton.

The problem? These maps are notoriously "blurry." Depending on which scientist draws the map or what mathematical "pen" they use, the map looks different. Because the maps aren't perfect, calculating the "chaos" (entropy) from them is like trying to guess the size of a forest by looking at a blurry photo of a single tree.

The Paper's Solution: Instead of using the blurry maps (PDFs), the author uses Structure Functions. Think of these as actual physical measurements—like measuring the actual temperature or volume of the crowd—rather than relying on a drawing of it. This makes the math much more "transparent" and reliable.

3. The Discovery: The "Scaling Effect"

The author compared his mathematical predictions against real-world data from a massive particle accelerator called HERA.

He found that his method—calculating entropy directly from the physical measurements—matched the experimental data almost perfectly. It’s like predicting how loud a party is just by measuring the vibration of the floor, and being right!

He also discovered something interesting about the "speed" of the chaos:

The "Higher Twist" Correction: At very low energies, the math gets a bit messy (like trying to hear a conversation in a room with a loud air conditioner). He added a "correction term" (a mathematical noise-canceler) to make the predictions even more accurate.
The Limit: He found that as you change the energy of the "magnifying glass," the amount of information you can extract changes in a very predictable, mathematical way.

4. Why does this matter? (The Future)

The paper concludes by looking toward the future. We are building even bigger "magnifying glasses" (colliders) called the EIC and the LHeC.

By using the formulas in this paper, scientists won't just be looking at "dots" in a machine; they will be able to use Information Theory to map out the fundamental "complexity" of the universe. It’s a way of moving from just seeing what is inside an atom to understanding how much information the universe is actually holding.

In short: The paper provides a better, clearer "thermometer" to measure the hidden complexity and "connectedness" inside a proton, moving away from blurry theoretical maps and toward solid, measurable reality.

Technical Summary: Entropy and DIS Structure Functions

Author: G.R. Boroun
Affiliation: Department of Physics, Razi University, Iran
Subject Area: High-Energy Physics (Quantum Chromodynamics / Deep Inelastic Scattering)

1. The Problem: The Observability of Entanglement Entropy

In Deep Inelastic Scattering (DIS), entanglement entropy is a theoretical tool used to probe the partonic structure (quarks and gluons) of the proton. Traditionally, entanglement entropy is calculated using Parton Distribution Functions (PDFs). However, PDFs are not directly observable physical quantities; they are scheme-dependent, subject to parametrization uncertainties, and require a factorization scale ( $\mu$ ). This creates a "theoretical tension" when trying to compare entropy predictions directly with experimental data, as different PDF fits can yield different results even if they describe the same physical observables.

2. Methodology: The Momentum-Space Approach

To resolve the ambiguity of scheme-dependent PDFs, the author proposes a method to define entanglement entropy directly in terms of observable DIS structure functions ( $F_2$ and $F_L$ ).

Inversion Technique: Instead of using PDFs as inputs, the author uses the mathematical inverse of the collinear factorization framework. By inverting the relations between structure functions and PDFs, the gluon distribution ($xg$) and the singlet distribution ( $x\Sigma$ ) are expressed entirely through $F_2$ , $F_L$ , and their derivatives (Eqs. 14–16).
Entropy Definition: The entanglement entropy $S(x, Q^2)$ is then derived as a function of these observable structure functions (Eq. 17).
Hadronic Multiplicity Scaling: To compare with experimental data regarding charged hadrons, the author applies a scaling factor of $2/3$ to the partonic entropy, accounting for the ratio of charged hadrons to the total hadron multiplicity (Eq. 23).
Higher-Twist (HT) Corrections: To improve accuracy at low- $x$ and low- $Q^2$ (where non-perturbative effects are significant), the author incorporates a phenomenological higher-twist term into the $F_2$ structure function (Eq. 25).
Kinematic Limit: The study examines the specific limit of high inelasticity ( $y \to 1$ ), where the minimum Bjorken $x$ is defined by $x_{min} = Q^2/s$ . At this point, the virtual photon polarization is transverse.

3. Key Contributions

Observable-Based Entropy: Developed a framework where entanglement entropy is a function of measurable structure functions, bypassing the need for unobservable PDF parametrizations.
Improved Precision at Low Scales: Demonstrated that adding higher-twist corrections significantly improves the description of entropy in the low- $Q^2$ and low- $x$ regimes.
Predictive Modeling for Future Colliders: Provided specific predictions for the entanglement entropy behavior at the upcoming Electron-Ion Collider (EIC) and the Large Hadron Electron Collider (LHeC).
Mathematical Refinement: Utilized Laplace transform techniques and the Block-Durand-Ha (BDH) parameterization to ensure the model aligns with the Froissart bound and experimental data.

4. Results

Alignment with H1 Data: The calculated entropy for charged hadrons (CE) shows excellent agreement with the H1 Collaboration's experimental data across a wide range of $x$ and $Q^2$ .
Quadratic vs. Linear Behavior: The results show a "considerable quadratic behavior" in the entropy relative to $x$ , which distinguishes this method from previous models (like HSS or HERAPDF) that exhibited more linear behavior.
Kinematic Trends:
- At the limit $x_{min} = Q^2/s$ , the entropy reaches a maximum (approximately 3 to 3.5) in the $20 < Q^2 \lesssim 100 \text{ GeV}^2$ range.
- The entropy exhibits a characteristic shape: increasing for $Q^2 < 20 \text{ GeV}^2$ , plateauing between $20$ and $100 \text{ GeV}^2$ , and decreasing for larger $Q^2$ .
Effective Intercept: The derivative of the entropy with respect to $\ln(1/x)$ reveals an "effective intercept" that governs the evolution of the entanglement, providing a new way to characterize the growth of partonic states.

5. Significance

This paper advances the field of "Information-Theoretic QCD" by bridging the gap between abstract quantum entanglement theory and concrete experimental measurement. By shifting the definition of entropy from unobservable PDFs to observable structure functions, the author provides a more transparent, unambiguous, and robust tool for probing the gluon-dominated structure of the proton. This methodology is particularly significant for interpreting data from next-generation high-energy colliders (EIC and LHeC), where understanding the entanglement of partons will be crucial for mapping the proton's internal landscape.